Introduction LetAdenote the class of functions of the form f(z) =z+ P∞ n=2anzn (1) which are analytic in the open unit disc U ={z : |z| <1}

December 2010

A GENERALIZED CLASS OF STARLIKE FUNCTIONS ASSOCIATED WITH THE WRIGHT HYPERGEOMETRIC FUNCTION

J. Dziok and G. Murugusundaramoorthy

Abstract. In terms of Wright’s generalized hypergeometric function we define some classes of analytic functions. The class generalize well known classes of starlike functions. Necessary and sufficient coefficient bounds are given for functions in this class. Further distortion bounds, extreme points and results on partial sums are investigated.

1. Introduction LetAdenote the class of functions of the form

f(z) =z+ P^∞

n=2anzⁿ (1)

which are analytic in the open unit disc U ={z : |z| <1}. We denote by S the subclass ofAconsisting functionsf which are univalent inU. Also we denote byT, the class of analytic functions with negative coefficients introduced by Silverman [14] consisting of functionsf of the form

f(z) =z− P^∞

n=2

anzⁿ (an≥0, n= 2,3, . . .;z∈U). (2) For functionsf ∈ Agiven by (1) andg(z)∈ Agiven by

g(z) =z+ P^∞

n=2bnzⁿ, z∈U,

we define the Hadamard product (or convolution) off and gby (f∗g)(z) =z+ P^∞

n=2

anbnzⁿ, z∈U.

Definition 1. Let k≥0, 0≤γ <1. A functionf ∈ Ais said to be in the classS(k, γ) if it satisfies the condition

Re

½zf⁰(z)

f(z) +kz²f⁰⁰(z) f(z)

¾

> γ, z∈U.

2010 AMS Subject Classification: 30C45.

Keywords and phrases: Analytic functions; starlike functions; coefficient estimates.

271

The classS(k, γ) was studied in [8] (see also [6, 9, 11]). In particular the class S(γ) :=S(0, γ)

is the well-known class of starlike functions.

For positive real parameters α1, A1. . . , αp, Ap and β1, B1. . . , βq, Bq (p, q ∈ N = 1,2,3, . . .) such that

1 + P^q

n=1

Bn− P^p

n=1

An≥0, (3)

the Wright’s generalized hypergeometric function [20]

pΨq[(α1, A1), . . . ,(αp, Ap); (β1, B1), . . . ,(βq, Bq);z] = pΨq[(αn, An)1,p; (βn, Bn)1,q;z]

is defined by

pΨq[(αt, At)1,p; (βt, Bt)1,q;z] = P^∞

n=0

{Q_p

t=0Γ (αt+nAt)}{Q_q

t=0Γ (βt+nBt)}⁻¹zⁿ n!, z ∈U. If p≤q+ 1, At = 1(t = 1, . . . , p) and Bt = 1(t = 1, . . . , q), we have the relationship:

ΘpΨq[(αn,1)1,p; (βn,1)1,q;z] = pFq(α1, . . . , αp; β1, . . . , βq;z), z∈U, (4) wherepFq(α1, . . . , αp; β1, . . . , βq;z) is the generalized hypergeometric function and

Θ = (Q_p

t=0Γ(αt))⁻¹(Q_q

t=0Γ(βt)). (5)

In [2] Dziok and Raina defined the linear operator by using Wright’s generalized hypergeometric function. Let

pφq[(αt, At)1,p; (βt, Bt)1,q;z] = Θz pΨq[(αt, At)1,p(βt, Bt)1,q;z], z∈U, and

W=W[(αn, An)1,p; (βn, Bn)1,q] :A→A be a linear operator defined by

Wf(z) :=z pφq[(αt, At)1,p; (βt, Bt)1,q;z]∗f(z), z∈U.

We observe that, forf of the form (1), we have Wf(z) =z+ P^∞

n=2

σn anzⁿ, z∈U, (6)

where

σn = Θ Γ(α1+A1(n−1)). . .Γ(αp+Ap(n−1)) (n−1)!Γ(β1+B1(n−1)). . .Γ(βq+Bq(n−1)) ,

and Θ is given by (5). In view of the relationship (4) the linear operator (6) includes the Dziok-Srivastava operator [3] (see [4]).

Corresponding to the familyS(γ, k) we define the classW_q^p(k, γ) of function f of the form (1) such that

Re

½z(Wf(z))⁰

Wf(z) +kz²(Wf(z))⁰⁰ Wf(z)

¾

> γ, z∈U. (7)

We also let

T W^p_q(k, γ) =T ∩ W_q^p(k, γ).

Further we define some subclasses of the classW_q^p(k, γ) as given below.

Suppose that: If At = 1 (t = 1, . . . , p) and Bt = 1(t = 1, . . . , q), p= 2 and q= 1 we have

(i)α1=δ+ 1 (δ >−1),α2= 1,β1= 1, then W(δ+ 1,1; 1)f(z)≡D^δf(z) := z

(1−z)^δ+1 ∗f(z)

is called Ruscheweyh derivative of orderδ(δ >−1)(see [12]). A function f ∈ Ais inRS(γ, k) if

Re

½z(D^δf(z))⁰

D^δf(z) +kz²(D^δf(z))⁰⁰ D^δf(z)

¾

> γ, z∈U, (8)

(ii)α1=a(a >0),α2= 1,β1=c(c >0), W(a,1;c)f(z)≡ L(a, c)f(z) :=

µ_∞ P

n=0 (a)n

(c)nzⁿ⁺¹

¶

∗f(z) called Carlson and Shaffer operator [1]. A functionf ∈ Ais in LS(γ, k) if

Re

½z(L(a, c)f(z))⁰

L(a, c)f(z) +kz²(L(a, c)f(z))⁰⁰ L(a, c)f(z)

¾

> γ, z∈U, (9) (iii)α1= 2 α2= 1,β1= 2−η,

W(2,1; 2−η)f(z)≡Ω^ηf(z) = Γ(2−η)z^ηD^η_zf(z)

where (η ∈ R;η 6= 2,3,4, . . .) the operator Ω^ηf(z) was introduced by Owa and Srivastava [19]. A functionf ∈ Ais inGS(γ, k) if

Re

½z(Ω^ηf(z))⁰

Ω^ηf(z) +kz²(Ω^ηf(z))⁰⁰ Ω^ηf(z)

¾

> γ, z∈U. (10)

In this paper we obtain a sufficient coefficient condition for functionsf given by (1) to be in the classW_q^p(k, γ) and we show that it is also necessary condition for functions to belong to the class. Distortion results and extreme points for functions inT W^p_q(k, γ) are obtained. Finally, we investigate partial sums for the classT W^p_q(k, γ).

2. Coefficients inequalities and distortions theorem First we obtain a sufficient condition for functions inW_q^p(k, γ).

Theorem 1. Let 0≤γ <1 and k≥0. Suppose also that f(z)∈ Ais given by(1). If

P∞ n=2

£kn²+n−kn−γ¤

σn|an| ≤1−γ, (11)

thenf ∈ W_q^p(k, γ).

Proof. If we put

P(z) =z(Wf(z))⁰

Wf(z) +kz²(Wf(z))⁰⁰

Wf(z) , z∈U, then it is sufficient to prove that

|P(z)−1|<1−γ, z∈U.

Indeed if f(z)≡z (z∈U), then we haveP(z)≡1 (z∈U). This implies that the desired in equality (11). If f(z)6= z (|z| =r < 1), then there exist a coefficient σnan6= 0 for somen≥2. It follows that

P∞ n=2

σn|an|rⁿ>0.

Further note that the sequencebn=kn²+n−kn−γis increasing. Therefore P∞

n=2

£kn²+n−kn−γ¤

σn|an|rⁿ >(1−γ)P^∞

n=2σn|an|rⁿ, which implies that P^∞

n=2σn|an|<1.

Thus by coefficient inequality (11), we obtain

|P(z)−1|=

¯¯

¯¯

¯¯

¯¯ P∞ n=2

(n−1)(nk+ 1)σnanzⁿ⁻¹ 1 + P^∞

n=2σnanzⁿ⁻¹

¯¯

¯¯

¯¯

¯¯

<

P∞ n=2

(n−1)(nk+ 1)σn|an| 1− P^∞

n=2σn|an|

= P∞ n=2

(n−1)(nk+ 1)σn|an| 1− P^∞

n=2

σn|an|

<

P∞ n=2

£kn²+n−kn−γ¤

σn|an| −(1−γ) P^∞

n=2

σn|an| 1− P^∞

n=2σn|an|

≤

(1−γ)−(1−γ) P^∞

n=2

σn|an| 1− P^∞

n=2

σn|an|

≤1−γ, z∈U.

Hence we obtain Re

½z(Wf(z))⁰

Wf(z) +kz²(Wf(z))⁰⁰ Wf(z)

¾

= Re(P(z))>1−(1−γ) =γ (z∈U).

That isf ∈ W_q^p(k, γ). This completes the proof.

In the next theorem, we show that the condition (11) is also necessary for functions from the classT W^p_q(k, γ).

Theorem 2. Let f be given by(2). Then the function f belongs to the class T W^p_q(k, γ)if and only if

P∞ n=2

£kn²+n−kn−γ¤

σ_na_n ≤1−γ. (12)

Proof. In view of Theorem 1 we need only to show thatf ∈ T W^p_q(k, γ) satisfies the coefficient inequality (11). Iff ∈ T W^p_q(k, γ) then the function

P(z) = z(Wf(z))⁰+kz²(Wf(z))⁰⁰

Wf(z) (z∈U) satisfies

Re{P(z)}> γ (z∈U).

This implies that

Wf(z) =z− P^∞

n=2

σnanzⁿ6= 0 (z∈U\ {0}).

Noting that ^Wf(r)_r is the real continuous function in the open interval (0,1) withf(0) = 1, we have

1− P^∞

n=2

σnanrⁿ⁻¹>0 (0< r <1). (13) Now

γ < P(r) = 1− P^∞

n=2

nσ_na_nrⁿ⁻¹−k P^∞

n=2

¡n²−n¢

σ_na_nrⁿ⁻¹ 1− P^∞

n=2

σnanrⁿ⁻¹ and consequently by (13) we obtain

P∞ n=2

£kn²+n−kn−γ¤

σnanrⁿ⁻¹<1−γ.

Lettingr→1, we get (12). This proves the converse part.

Corollary 1. If a functionf of the form(2)belongs to the classT W^p_q(k, γ), then

an≤ 1−γ

[kn²+n−kn−γ]σn, n= 2,3, . . . .

The equality holds for the functions hn(z) =z− 1−γ

[kn²+n−kn−γ]σnzⁿ, z∈U, n= 2,3, . . . . (14) Next we obtain the distortion bounds for functions belonging to the class T W^p_q(k, γ).

Theorem 3. Let f be in the classT W^p_q(k, γ), |z|=r <1. If the sequence

©£kn²+n−kn−γ¤ σn

ª_∞

n=2

is nondecreasing, then r− 1−γ

(kγ+ 2)σ2r²≤ |f(z)| ≤r+ 1−γ

(2k−γ+ 2)σ2r². (15) If the sequence

nkn²+n−kn−γ

n σn

o_∞

n=2 is nondecreasing, then 1− 2(1−γ)

(2k−γ+ 2)σ2r≤ |f⁰(z)| ≤1 + 2(1−γ)

(2k−γ+ 2)σ2r. (16) The result is sharp. The extremal function is the functionh2 of the form(14).

Proof. Sincef ∈ T W^p_q(k, γ), we apply Theorem 2 to obtain (2k−γ+ 2)σ2

P∞

n=2|an| ≤ P^∞

n=2

£kn²+n−kn−γ¤

σn|an| ≤1−γ.

Thus

|f(z)| ≤ |z|+|z|² P^∞

n=2

|an| ≤r+ 1−γ

(2k−γ+ 2)σ2r². Also we have

|f(z)| ≥ |z| − |z|² P^∞

n=2|an| ≥r− 1−γ (2k−γ+ 2)σ2r² and (15) follows. In similar manner forf⁰, the inequalities

|f⁰(z)| ≤1 + P^∞

n=2

n|an||z|ⁿ⁻¹≤1 +|z| P^∞

n=2

nan

and P∞

n=2

n|a_n| ≤ 2(1−γ) (2k−γ+ 2)σ2

lead to (16). This completes the proof.

Corollary 2. Letf be in the classT W^p_q(k, γ), |z|=r <1. If

p > q, αq+1≥1, αj ≥βj and Aj≥Bj (j= 2, . . . , q), (17) then the assertions(15),(16)holds true.

Proof. From (17) we have that the sequences

©£kn²+n−kn−γ¤ σn

ª∞ n=2 and

½kn²+n−kn−γ

n σn

¾∞ n=2

are nondecreasing. Thus, by Theorem 3, we have Corollary 2.

Theorem 4. Let h1(z) = z andhn be defined by(14). Then f ∈ T W^p_q(k, γ) if and only if f can be expressed in the form

f(z) = P^∞

n=1

µnhn(z), µn≥0and P^∞

n=1

µn= 1. (18)

Proof. If a functionf is of the form (18), then f(z) = P^∞

n=2

£kn²+n−kn−γ¤

σn 1−γ

[kn²+n−kn−γ]σnµn

= P^∞

n=2

µn(1−γ) = (1−µ1)(1−γ)≤1−γ.

Hencef ∈ T W^p_q(k, γ).

Conversely, iffis in the classT W^p_q(k, γ), then we may setµn= [^{kn2+n−kn−γ}]^σn

1−γ ,

n≥2 andµ1= 1− P^∞

n=2µn. Then the function f is of the form (18) and this com- pletes the proof.

3. Partial sums

For a functionf ∈A given by (1) Silverman [15] and Silvia [18] investigated the partial sumsf1and fmdefined by

f1(z) =z; and fm(z) =z+ P^m

n=2

anzⁿ, m= 2,3. . . . (19) We consider in this section partial sums of functions in the class T W^p_q(k, γ) and obtain sharp lower bounds for the ratios of real part off tof_mandf⁰ tof_m⁰ .

Theorem 5. Let a function f of the form (2) belong to the classT W^p_q(k, γ) and assume(17). Then

Re

½ f(z) fm(z)

¾

≥1− 1 dm+1

, z∈U, m∈N (20)

and

Re

½f_m(z) f(z)

¾

≥ d_m+1

1 +dm+1, z ∈U, m∈N, (21) where

dn :=kn²+n−kn−γ

1−γ σn. (22)

Proof. By (17) it is not difficult to verify that

d_n+1> d_n>1, n= 2,3. . . . (23) Thus by Theorem 2 we have

Pm n=2

an+dm+1

P∞ n=m+1

an≤ P^∞

n=2

dnan≤1. (24)

Setting

g(z) =dm+1

½ f(z) fm(z)−

µ 1− 1

dm+1

¶¾

= 1 + dm+1

P∞ n=m+1

anzⁿ⁻¹ 1 + P^m

n=2

anzⁿ⁻¹

, (25)

it suffices to show that Reg(z)≥0, z∈U. Applying (24), we find that

¯¯

¯¯g(z)−1 g(z) + 1

¯¯

¯¯≤

d_m+1 P^∞

n=m+1

|a_n| 2−2 Pⁿ

n=2|an| −dm+1

P∞ n=m+1|an|

≤1, z∈U,

which readily yields the assertion (20) of Theorem 5. In order to see that f(z) =z+z^m+1

dm+1

, z∈U, (26)

gives the sharp result, we observe that forz=re^iπ/m we have f(z)

fm(z)= 1 + z^m dm+1

z→1⁻

−→ 1− 1 dm+1. Similarly, if we take

h(z) = (1 +dm+1)

½fm(z)

f(z) − dm+1

1 +dm+1

¾

= 1−

(1 +dn+1) P^∞

n=m+1

anzⁿ⁻¹ 1 + P^∞

n=2

anzⁿ⁻¹

, z∈U,

and making use of (24), we can deduce that

¯¯

¯¯h(z)−1 h(z) + 1

¯¯

¯¯≤

(1 +dm+1) P^∞

n=m+1

|an| 2−2 P^m

n=2

|an| −(1−dm+1) P^∞

n=m+1

|an|

≤1, z∈U,

which leads us immediately to the assertion (21) of Theorem 5. The bound in (21) is sharp for eachm∈N with the extremal functionf given by (26), and the proof is complete.

Theorem 6. Let a function f of the form (1) belong to the classT W^p_q(k, γ) and assume(17). Then

Re

½f⁰(z) f_m⁰ (z)

¾

≥1−m+ 1 dm+1

(27)

and

Re

½f_m⁰ (z) f⁰(z)

¾

≥ dm+1

m+ 1 +dm+1, (28)

wheredm is defined by(22).

Proof. By setting

g(z) =dm+1

½f⁰(z) f_m⁰ (z)−

µ

1−m+ 1 dm+1

¶¾

, z∈U, and

h(z) = [(m+ 1) +dm+1]

½f_m⁰ (z)

f⁰(z) − dm+1

m+ 1 +dm+1

¾

, z∈U, the proof is analogous to that of Theorem 5, and we omit the details.

4. Integral transform and integral means

First we prove that the classT W^p_q(k, γ) is closed under some integral trans- forms.

Forf ∈ Awe define the Komatu operator [10]

Vδ,c(f)(z) =(c+ 1)^δ Γ (δ)

Z ₁

0

t^c−1 µ

log1 t

¶_δ−1

f(tz)dt, c >−1, δ≥0.

In particular, forδ= 1 we obtain well-known the Bernardi operator.

A simple calculation gives

Vδ,c(f)(z) =z− P^∞

n=2

µc+ 1 c+n

¶_δ anzⁿ. First we show that the classT W^p_q(k, γ) is closed underVδ,c(f)(z).

Theorem 7. Let f(z)∈ T W^p_q(k, γ). ThenVδ,c(f)(z)∈ T W^p_q(k, γ).

Proof. We need to prove that P∞ n=2

Φ(λ, γ, k, n) (1−γ)

³c+1 c+n

´_δ

an≤1, (29)

where

Φ(γ, k, n) = [kn²+n−kn−γ]σn. (30) On the other hand by Theorem 2,f ∈ T W^p_q(k, γ) if and only if

P∞ n=2

Φ(λ, γ, k, n) (1−γ) an ≤1,

where Φ(γ, k, n) is defined in (30). Since _c+n^c+1 <1, then (29) holds and the proof is complete.

Theorem 8. Let f ∈ T W^p_q(k, γ). Then Vδ,c(f)(z) is starlike of order ξ, 0≤ξ <1in |z|< R1, where

R1= inf

n

"µ c+n c+ 1

¶_δ

(1−ξ)Φ(γ, k, n) (n−ξ)(1−γ)

# ¹

n−1

(n≥2), whereΦ(γ, k, n)is defined in (30).

Proof. It is sufficient to prove

¯¯

¯¯z(Vδ,c(f)(z))⁰ Vδ,c(f)(z) −1

¯¯

¯¯<1−ξ, |z|< R1. (31) For the left hand side of (30) we have,

¯¯

¯¯z(Vδ,c(f)(z))⁰ Vδ,c(f)(z) −1

¯¯

¯¯=

¯¯

¯¯

¯¯

¯¯

¯ P∞ n=2(1−n)

µc+ 1 c+n

¶δ

anzⁿ⁻¹ 1− P^∞

n=2

µc+ 1 c+n

¶_δ a_nzⁿ⁻¹

¯¯

¯¯

¯¯

¯¯

¯

≤ P∞ n=2(1−n)

³c+1 c+n

´_δ

an|z|ⁿ⁻¹ 1− P^∞

n=2

µc+ 1 c+n

¶_δ

an|z|ⁿ⁻¹ .

The last expression is less than 1−ξif

|z|ⁿ⁻¹<

µc+n c+ 1

¶_δ

(1−ξ)Φ(γ, k, n) (n−ξ)(1−γ) . Therefore, the proof is complete.

Using the fact thatf(z) is convex if and only ifzf⁰(z) is starlike, we obtain the following.

Theorem 9. Letf ∈ T W^p_q(k, γ). ThenVδ,c(f)(z)is convex of order0≤ξ <1 in|z|< R2 where

R2= inf

n

"µ c+n c+ 1

¶_δ

(1−ξ)Φ(γ, k, n) n(n−ξ)(1−γ)

# ¹

n−1

(n≥2), whereΦ(λ, γ, k,2) is defined in (30).

In [14], Silverman found that the function f2(z) = z− ^z₂² is often extremal over the familyT. He applied this function to resolve his integral means inequality, conjectured in [16] and settled in [17], that

Z _2π

0

¯¯f(re^iθ)¯

¯^η dθ≤ Z _2π

0

¯¯f2(re^iθ)¯

¯^η dθ,

for all f ∈T, η >0 and 0< r < 1. In [17], he also proved his conjecture for the subclassesT^∗(γ) andC(γ) ofT.

Now, we prove the Silverman’s conjecture for the functions in the family T W^p_q(k, γ).

Lemma 1. [7]If the functions f andg are analytic in U with g≺f, then for η >0, and0< r <1,

Z _2π

0

¯¯g(re^iθ)¯

¯^η dθ≤ Z _2π

0

¯¯f(re^iθ)¯

¯^η dθ. (32)

Applying Lemma 1, Theorem 2 and Theorem 4 we prove the following result.

Theorem 10. Supposef ∈ T W^p_q(k, γ),η >0,0≤λ≤1 andf2(z)is defined by

f2(z) =z− 1−γ Φ(γ, k,2)z²,

whereΦ(γ, k,2) is defined in(30). Then forz=re^iθ,0< r <1, we have Z _2π

0

|f(z)|^η dθ≤ Z _2π

0

|f2(z)|^η dθ. (33)

Proof. By (2), it is equivalent to prove that Z _2π

0

¯¯

¯¯1− P^∞

n=2

anzⁿ⁻¹

¯¯

¯¯

η

dθ≤ Z _2π

0

¯¯

¯¯1− (1−γ) Φ(λ, γ, k,2)z

¯¯

¯¯

η

dθ.

By Lemma 1, it suffices to show that 1− P^∞

n=2

anzⁿ⁻¹≺1− 1−γ Φ(λ, γ, k,2)z.

Setting

1− P^∞

n=2

anzⁿ⁻¹= 1− 1−γ

Φ(λ, γ, k,2)w(z), (34)

and using (11), we obtain

|w(z)|=

¯¯

¯¯ P∞ n=2

Φ(γ,k,n) 1−γ anzⁿ⁻¹

¯¯

¯¯≤ |z| P^∞

n=2

Φ(γ, k, n)

1−γ an≤ |z|.

This completes the proof.

5. Neighbourhood results

In this section we discuss neighbourhood results of the classT W^p_q(k, γ). Fol- lowing [5, 13], we define theδ-neighbourhood of functionf(z)∈T by

Nδ(f) :=

½

h∈T :h(z) =z− P^∞

n=2

dnzⁿ and P^∞

n=2

n|an−dn| ≤δ

¾

. (35)

Particulary for the identity functione(z) =z, we have Nδ(e) :=

½

h∈T :g(z) =z− P^∞

n=2dnzⁿ and P^∞

n=2n|dn| ≤δ

¾

. (36)

Theorem 11. If

δ:= 2(1−γ)

Φ(λ, γ, k,2), (37)

whereΦ(λ, γ, k,2) is defined in (30), thenT W^p_q(k, γ)⊂Nδ(e).

Proof. Forf ∈ T W^p_q(k, γ), Theorem 2 immediately yields Φ(λ, γ, k,2) P^∞

n=2

an ≤ 1−γ, so that

P∞ n=2

an ≤ (1−γ)

Φ(λ, γ, k,2). (38)

On the other hand, from (11) and (38) that σ2(1 +λ) P^∞

n=2

nan≤(1−γ)−(1 +λ)(k−γ)σ2

P∞ n=2

an

≤(1−γ)−(1 +λ)(k−γ)σ2 (1−γ)

Φ(λ, γ, k,2) ≤ 2(1−γ) [2 +k−γ], that is

P∞ n=2

na_n ≤ 2(1−γ)

Φ(λ, γ, k,2) :=δ (39)

which, in view of the definition (36) proves Theorem 11.

Remark. We observe that, if we specialize the parameters of the class T W^p_q(k, γ), we obtain the analogous results for the classesRS(γ, k), LS(γ, k) and GS(γ, k) defined in this paper.

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(received 18.07.2009; in revised form 07.11.2009)

J. Dziok, Institute of Mathematics, University of Rzesz´ow ul. Rejtana 16A, PL-35-310 Rzesz´ow, Poland E-mail:[email protected]

G. Murugusundaramoorthy, School of Science and Humanities, V I T University, Vellore - 632014, T.N., India

E-mail:[email protected]